Problem: Simplify and expand the following expression: $ \dfrac{n}{n - 6}-\dfrac{n}{n - 4} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(n - 6)(n - 4)$ Multiply the first term by $\dfrac{n - 4}{n - 4}$ $ \begin{align*} \dfrac{n}{n - 6} \times \dfrac{n - 4}{n - 4} & = \dfrac{(n)(n - 4)}{(n - 6)(n - 4)} \\ & = \dfrac{n^2 - 4n}{(n - 6)(n - 4)}\end{align*} $ Multiply the second term by $\dfrac{n - 6}{n - 6}$ $ \begin{align*} \dfrac{n}{n - 4} \times \dfrac{n - 6}{n - 6} & = \dfrac{(n)(n - 6)}{(n - 4)(n - 6)} \\ & = \dfrac{n^2 - 6n}{(n - 4)(n - 6)}\end{align*} $ Now we have: $ = \dfrac{n^2 - 4n}{(n - 6)(n - 4)} - \dfrac{n^2 - 6n}{(n - 4)(n - 6)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{n^2 - 4n - (n^2 - 6n)}{(n - 6)(n - 4)} $ $ = \dfrac{n^2 - 4n - n^2 + 6n}{(n - 6)(n - 4)} $ $ = \dfrac{2n}{(n - 6)(n - 4)}$ Expand the denominator: $ = \dfrac{2n}{n^2 - 10n + 24}$